The Basic Catches

There are three basic catches: the Alligator Catch, the Two Hand Catch and the One Hand Catch.

The Alligator Catch is the easist catch. Put the disc down. Put your hands together like you are clapping.
Now, turn both hands sideways so you’re clapping up and down instead of side by side. Sort of like an alligator’s jaws are opening.
If someone throws a disc to you, get in this position. When the disc reaches the alligator’s jaws, clamp down and catch.
If you don’t have a friend to practice with, put the disc in the alligator’s jaws and practice clamping down on the disc by opening and closing the alligator’s jaws.

For the Two Hand Catch, hold both hands at chest level with your thumbs lower than your fingers.
Open and close your thumb/fingers like you are doing a shadow puppet show. When you’re friend throws you the disc, clamp down both hands just before the disc hits your hand.
Don’t wait too long or the disc will hit your hand, bounce away and hit the ground with a big thud.

The One Hand Catch is the key to all the pro catches. Almost every trick catch involves catching with only one hand. There are two hand positions for the basic one hand catch, one
for high throws and one for low throws.

If a throw flies toward you at chest level or higher, your thumb should be below your other fingers. Your palm should be facing down or out.
If a throw flies toward you lower than chest level, your thumb should be above your fingers. Your palm should be facing upward.
The video below (108kb) shows the palm-up (low throw) one handed catches: